To determine The displacement, when d 2 x d t 2 is the acceleration. Explanation Theorem used: If r 1 = α + i β and r 1 = α − i β are the two complex roots of the auxiliary equation a r 2 + b r + c = 0 , then y = e α x ( c 1 cos β x + c 2 sin β x ) is the general solution of a y ″ + b y ′ + c y = 0 . Calculation: Simple harmonic motion may be defined as motion in a straight line for which the acceleration is proportional to the displacement and in the opposite direction. d 2 x d t 2 = − k 2 x To write the above equation if the form: D 2 x + k 2 x = 0 The auxiliary equation is: D 2 + k 2 = 0 D 2 = − k 2 D = ± j k So, the general solution to the above differential equation is x = e 0 t ( c 1 sin k t + c 2 cos k t ) = c 1 sin k t + c 2 cos k t This solution indicates an oscillating motion, which is known to be the case. If, for example, k = 4 and to know that x = 0 and D x = 2 for t = 0 . x = c 1 sin 4 t + c 2 cos 4 t Substitute the initial conditions x = 0 when t = 0 in x = c 1 sin k t + c 2 cos k t

Student Solutions Manual For Basic Technical Mathematics And Basic Technical Mathematics With Calculus

11th Edition

ISBN: 9780134434636

Author: Allyn J. Washington, Richard Evans

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 31.10, Problem 1PE

To determine

The displacement, when d2xdt2 is the acceleration.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 31.9, Problem 40E

Chapter 31.10, Problem 1E

Students have asked these similar questions

7. Show that for R sufficiently large, the polynomial P(z) in Example 3, Sec. 5, satisfies the inequality |P(z)| R. Suggestion: Observe that there is a positive number R such that the modulus of each quotient in inequality (9), Sec. 5, is less than |an|/n when |z| > R.

9. Establish the identity 1- 1+z+z² + 2n+1 ... +z" = 1- z (z1) and then use it to derive Lagrange's trigonometric identity: 1 1+ cos cos 20 +... + cos no = + 2 sin[(2n+1)0/2] 2 sin(0/2) (0 < 0 < 2л). Suggestion: As for the first identity, write S = 1+z+z² +...+z" and consider the difference S - zS. To derive the second identity, write z = eie in the first one.

8. Prove that two nonzero complex numbers z₁ and Z2 have the same moduli if and only if there are complex numbers c₁ and c₂ such that Z₁ = c₁C2 and Z2 = c1c2. Suggestion: Note that (i≤ exp (101+0) exp (01-02) and [see Exercise 2(b)] 2 02 Ꮎ - = = exp(i01) exp(101+0) exp (i 01 - 02 ) = exp(102). i 2 2

Chapter 31 Solutions

Student Solutions Manual For Basic Technical Mathematics And Basic Technical Mathematics With Calculus

Ch. 31.1 - Show that is a solution of . Is it the general...Ch. 31.1 - Prob. 1E Ch. 31.1 - In Exercises 1 and 2, show that the indicated...Ch. 31.1 - In Exercises 3–6, determine whether the given...Ch. 31.1 - Prob. 4E Ch. 31.1 - In Exercises 3–6, determine whether the given...Ch. 31.1 - Prob. 6E Ch. 31.1 - In Exercises 7–10, show that each function y =...Ch. 31.1 - Prob. 8E Ch. 31.1 - In Exercises 7–10, show that each function y =...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.